Fast Detection of the Number of Signals Based on Ritz Values: A Noise-Power-Aided Approach

Detecting the number of signals (NOS) is afundamental problem in array processing. However, most of the existing methods have a computational complexity of
$O(M^3)$
, where
$M$
is the number of sensors. In this article, we propose a fast detection algorithm with a complexity of
$O(DM^2)$
, where
$D$
is the NOS. Different from the conventional approaches that depend on the eigenvalues of the sample covariance matrix (SCM), the proposed method is based on the Ritz values—the eigenvalues of the projected SCM in the Krylov subspace. We first present a scheme to construct the Krylov subspace via the Lanczos tridiagonalization. Then, we prove that the first
$D$
Ritz values are equal to the
$D$
signal eigenvalues. Furthermore, if the Krylov subspace has a dimension greater than
$D$
, the minimum Ritz value (MRV) converges to the noise power. Therefore, we can detect the NOS by comparing the MRV with a threshold determined by the noise power. Because the proposed algorithm does not perform eigenvalue decomposition, it has a simple implementation structure. Moreover, by using the Sylvester’s law of inertia, the Ritz values do not need to be computed either. Simulation results show that the proposed algorithm outperforms the existing methods under the conditions of small sample size and low signal-to-noise ratio.